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Asymptotic Approximation of high order Wavepackets

by R. Bourquin and V. Gradinaru

(Report number 2016-52)

Abstract
We demonstrate and analyze the failure of the three-term recursion on the evaluation of Hermite functions for important parameter and argument values. Asymptotic expansions inspire a solution to this problem. We explicitly develop the necessary fomulae in detail and implement an algorithm realizing this solution. The result is applicable to a wide range of input parameter values. The main goal is now an application to Hagedorn wavepackets in one dimension. We can improve the robustness of wavepacket based spectral methods as it becomes possible to evaluate wavepackets of much higher order. The simple example of an overlap matrix computation is shown where we can get rid of any erratic behavior.

Keywords: Hermite function Three-term recursion Asymptotic series expansion Airy function Special functions Fast evaluation algorithms Semiclassical wavepackets

@Techreport{BG16_689,
  author = {R. Bourquin and V. Gradinaru},
  title = {Asymptotic Approximation of high order Wavepackets},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-52},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-52.pdf },
  year = {2016}
}

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